u“- '- .4- ' ".q_'. ‘ ‘ 1,;1 i; M ‘l N H ]' Illzl I . | WI M ll 1 J l , (l YHEORY OF HEAT CONDUCTION AND CONVECTION FROM A HOT VERTICAL CYLINDER ‘YHESES PUB THE DEGREE OF M. A. Lawrence Duncan Childs 1931 . , . . v . 7 .nnv u o . L _ - . , , . u . min-1.? . u ur .) \ . I . . . a .1" ...Y :1 ...W....av.m¢..J.!c.\. . . .. . . .. . . . . .J.‘ o u r l . -10 . . - . ........rr} -. AW..V in... n}: e .. . .tpdfl. n.. .tJ... FL»... $33... a .. . . . .. .. . ....,........§. 5 a I’MJ «NU? .. %n‘.l~.A,.A&1r)\.waI‘-‘Wrazw u. . , 4 . .. WW3 Minn ..¢H.r... rU-fihunflrv. 4 u. :MNAWH. .. ‘3‘“. FruUle. -u; . . . .. , .. .9: L x... . . . . . .. . . -.o o P.9I ”if... PF. my. 3. VAL... «f‘ ANNA...” «Wflfwfi1flwlmwf buy... a»... .. ”I HM. .. no- N ....;: .4 ma:.JMHG.r. .JZVflwu. HJui: . 3.... _. a .. .....v41...u . LP..MA/r.).1. .o 1%3. .ADVLDLI—OAJIQMII. L»...J.l.n'./1 PEA. VI AAA KIWI» .. 4 I xllb ‘, I A] n .u .. .n. {- \ ~... CAI MOCLLQJZW A 2 \{lf'rCLiku .5. A w A 0954 A A A A A A A AA A A A A AIL A A A A A A E UNIVERSITY LIBRARIE I AAAj AA .r MMJH ‘A AA A] 31293 01701 A "AA AA A A AA‘ A MICHIGAN STAT ’ LIBRARY Michigan State Unlversi .. ‘.A. , - .... (1...... Irwv‘rvi.mum-uap . . PLACE IN RETURN BOX to remove th is checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. THEORY OF HEAT CONDUCTION AND CONVECTION FROM A HOT VERTICAL CYLINDER A Theeie Submitted to the Faculty of Michigan State College of Agriculture and Applied Science In partial fulfillment of the requirements for the degree of [ester of Arte by Lawrence Duncan Childe 1931 To Doctor lilliam Scribner Kimball without whoee eug— geetione and aid this theeie would not have been com. pleted. 103455 __ 1 -_ I. INTRODUCTION. GENERAL HEAT LAWS The purpose of this investigation is to extend the work of a recent paper * on heat conduction and convection from the plane case to that of a cylinder. The treatment is the same as that in the above mentioned paper, with modifications and restrictions to adapt-it to the cylindrical case. This in brief, is as follows: A cylinder heated to the temperature of boiling water stands vertically in an atmosphere of air at ordinary room temperature. Surrounding this cylinder, in its immediate neighborhood, is a thin film thru which heat flows by con- duction and convection. The development and discussion of the laws by which heat flows thru this film comprises this paper. Two empirical laws ‘* and a simplified hydrodynamics equation make possible the present treatment. LAI I. The locus of the maxima of the velocity curves is an isothermal surface whose temperature is the mean of the temperatures of the hot plate and the ambient air. LAW 11. Half the heat is ccnvected up away inside the film and half outside. This thin film is bounded by the cylinder and the isothermal surface defined by Law I. * l.8.ximba11 and l.J.King “Theory of Heat Conduction and Convection from a Hot Vertical Plate". Unpublished - Presented at the meeting of the American Mathematical Society April 4, 1931. “ Kimball and King (Same as above) p. 3 -- 3 -- L Cy/ma/er I r‘ ' n I I“ —---~ in” \ \ \ \ \ r \\ I I \ l \ \\ \ \ \ \\ \\ \ . \\ \x \\ \~ \~ R..— r __.).4 rdC/li/S Figure 1. Since the gas in this thin film is transporting heat by convection as well as by conduction, it is in motion. . Hence there are viscous and buoyancy forces which must be taken into consideration. In order to fully appreciate the effect of these viscous forces in a gas and the relation existing between buoyancy and the rate of doing work, let us state Kimball's Theorems *. THEORII I. The rate of doing work per unit volume by the buoyancy is proportional to the rate of heat transfer thru unit area. T3103]! 11. The viscous forces operating in a gas in a steady state form a mechanical couple, and hence their sum is always zero. Viscosity's sole mechanical effect is that of a couple: it passes on the reactions from place to place within the gas or to the walls of the container together ‘ Kimball and King p.4 -- 3 -- with a corresponding torque. There is also the illuminating corollary: The total buoyancy is always balanced by inertia effects plus a down- ward pull by the walls of the container which are together equal and ppposite to the viscous drag. These laws and theorems must be kept in mind and incor- ‘ggiated in the present treatment together with the funda- mental equations of heat *. Also the Langmuir film theory" is a convenient check upon this investigation. II. THE FILM THICKNESS; THE TEMPERATURE EXPRESSION If we denote the temperature of the gas at a point within the film by T , the temperature of the cylinder by T,, the radius of the cylinder by r the distance from the 0’ axis of the cylinder to the point by r , and the height of the point above the plane of the base by y , (see figure 1) we may form an empirical temperature expression (1 - —- /+é _.,,,)/ L ) T" 7; a/ 6 07f ’ where a, b, _and c1 are constants, two of which are eval- uated by the present treatment. If we use the symbol T", for the temperature of the film boundry, and To for the temperature of the ambient air, we have by Law 1 7"4- 7' 7- : I O ’4 . (a) m 2 Substituting this expression for temperature in (1) we ‘ J.G.Ooffin 'Vectcr Analysis“ pp. 104 - 116 '* Phys. Rev., 34, 401 (1913) -- 4 -_ obtain the relation 3 r = 77-7; ( ) lo; I; 24 {/*be-49) where r; is the radius of the film boundry. Now from (3) it is readily seen that 7,-7.3 (4) a. -- i; e 240W") which gives the film thickness sz; _ 2 1?? (5) 6.4.: - 6/6: an... )__,/ This shows that the film thickness increases with the height y. III. THE FORGE EQUATIOI Inside the film the empirical temperature expression (1) satisfies the differential equation 7‘ 7 2/:607 ’ (e) —K/—,—.Lj-J;{r2é7:)+ :77" j. y Y . where K is the coefficient of conductivity and (‘= 0‘ K This equation differs from the simplified classical hydro- dynamic equaticn for combined conduction and convection only in that C : «K , approximately a constant, instead of the convection energy flux density. This latter is given by ('3;an where k is Boltzmann's constant, n is the molecular concentration, and v is the convection velocity, and it varies from mere at the hot cylinder to a maximum at the film boundry. However this constancy of 0 seems to be experimentally Justified even though the theory here is somewhat incomplete 9. ‘ Kimball and King p. 3 -- 5 -- It may be easily seen that for a constant value of y the left member of (6) vanishes. This indicates pure con- duction outward from the cylinder thru the film for a constant height with no loss of heat on route. This is a refinement of the Langmuir film concept, a theory which represents resultant effects accurately in terms of pure conduction thru practically stationary gas in an equivalent film of constant thickness surrounding a hot body. lithin the film the fundamental equations of hydrodynamics are simplified by neglecting inertia effects, second order velocity effects, horizontal velocities, and by using the gas law. The downward viscous force I on an annular sur- face of dimensions ¢3fl7‘x1 will then be (see figure 3) m F= —Z1r»7r'07 i—V , where v7 is the coefficient of viscosity of the gas and v is its convection velocity. The difference between the force outside the ring ang the force inside is a resultant downward force which is the viscous drag ‘. This is given by (8) dF= —Z7TI7/ ”fl JV dr. If there is no acceleration, this force must be balanced by the buoyancy. Let m be the mass of a molecule of the gas g the acceleration due to gravity, n the molecular density of the hot gas and no that of the caller gas. Then mgn is the weight in absolute units per unit volume of hot gas and mgno the corresponding weight of the displaced unit volume of the cooler gas. From Archimedes principle, the buoyant ‘ Leigh Page 'Introduction to Theoretical PhysicsI p. 330 -- 6 -- force per unit volume is (9) ”79% — ”Vim M9 ”3-4. ‘3 I 1EEEDJr zuwfl‘ L_———-—~————'—_""'/ r Figure 2. The condition that the buoyancy balance the drag is therefore 0) oh/ - 21/77 ”fig; 1/“ - ermg/Igdfi/‘dr, from which we obtain the fundamental force equation (10) _ .. Li/réjem ”47 7/“ Jr /r i/” j It is illuminating to compare this fundamental force equation with the classical laws of Poiseuille, for viscous flow in capillary tubes, and LaPlace for pressure distri- bution in isothermal atmosphere. For the first case we have simply to replace the right band member of (10) by its equivalent in terms of pressure, that is, .baV/Qg-lgj -=‘:L/Qg 7gb) = cabjfiavf; , "' 4 _‘. .7“ 5“. -- 7 -- we now integrate the resulting expression and obtain a formula for velocity. The volume V of fluid passing any cross section in unit time is found to be given by the relation 5 4 fTC (11) V: ZW/VI’C/l' = 87! (fl ‘/9) ' a Formula (11) is the well known formula of Poiseuille‘. For the second case we assume that there is no viscosity, hence the left hand mamber of (10) is zero. By holding T constant and using the gas 1aw*' (12) p = n/rT , Instead 0/ #0) we have trcmzflnti 67= l137/7g-Iz);£5-I7g7h7aé/ = aéo' , c/ .- My,” 0/9 ‘ /7 k 7‘ (13) M /b = /3 <3 k7‘ ls have then the La Place law for pressure distribution"! Thus the fundamental force equation (10) is perhaps the simplest combination of the two classical formulas.cf Poissuille and La Place. ‘ L.B.Loeb 'Kinetic Theory of Gases" p. 843 ** Leigh Page p. 294 '** Hertzfeld 'Kinetische Theorie der warme' p. 81 -- 3 -- IV. CONVECTIOI VELOCITY With the application of the gas law (12) the force equation (10) becomes (14’ 7771’s '43me 7;"? :77; 7%) For simplicity, let us make the following definitions: x=/0 J: ' X” -/ ILL” - (15) 9,; J I fly)? :QAa/Héé/ ,flvm/y, Xo:/0.7;/f::0 Equation (14) may then be written in the simplified form 2 (16) —7 J y x mg? T—7; ,jy’ -. c. IfZ; // 7‘ /)’ and the temperature expression (1) becomes (17) 7“ 77 - m/néf") . By substituting for T in (16) its value as given by (17) we obtain a function of x . If we expand this as a power series in x using no terms greater than second order and integrate using law I to determine the constants of inte- gration, that 13.5%5=6’ when x ~xh1, we obtain as the result of the first integration the expression (18) via £2141. /7,*-—7/)r»x,,,+x’.x,,f+3é.{xixfl/ -,:§Loa/%¢?fly {gér1c§f23&:6/:j%%42?7nfiéflflyii&;{/z Integrating again using the fact that v= 0 at r= r , or in terms of the variable x we have from (15) that v==0 when x==0, we obtain for the convection velocity Z (19) = A4.“ ’"2 L‘C/ Hi +5— —§_"_7«;/{i:4_13- £14 V 74-70-71- 24“) 24a! .19an X Z 3 é ’ _ / / ,4?” _ "- 7: at!) wherein w-(zv‘zm’ 7/ , 4- 7‘4” :27? ’ and .5: 7-7—2973].— 13— azwz‘, By Law I the maximum convection velocity v”,occurs at the outer boundry of the film, that is, where x==xnfl hence from (19) we readily obtain J (20) v -_- ems/27H") p. =74- + see) . m 57*7zta‘w 3“” 2.... V. TRANSFER OF HEAT; TH! LANGMUIR FILM. The heat conduction per unit area*at the hot cylinder is r _ (21 =-—4~—/ =/[é./+éc~’ ) ? fldr‘ r4; (I; / c j the right hand expression being obtained by substituting forégzy its value obtained from the differentiation of the reg temperature expression (1). The total heat ccnvected away from a cylinder of height L is L I -4 / ml (33) 62 = errgyd’y = 27752 /r///+55 Vé/y = Zflaf/Z’f/I-C’U/I low according to Law II we may write (23) Q.- Z/raxf/Z = Z/faf—é//—€wy Z q ‘ Ooffin p. 104 -- 10 -- The heat transferred to the gas beyond the film is Zlm KL and Zfla/(qéflrd/ is the excess supply of heat at the lower part of the film due to the larger temperature gradient there. This excess heat is ccnvected up thru the film. Although the theory involved is somewhat incomplete, the fact repre- sented by (23) is justified experimentallyt. Is may interpret this equivalence of 2a- and Zvaxfl by reliance upon the Langmuir film concept, a theory which shows that actual rates of heat transfer, heat transfer coefficients, and so forth, are exactly what they would be if heat were transported away from a hot body by pure con- duction across a film of stationary gas and of constant thickness. For such a case, if Q is the total heat flow for pure conduction across any cylindrical surface within the Langmuir film at r distance from the axis, then Integrating between the temperature limits T, and To , and expressing the boundry of the Langmuir film by r;, we have (25) 0= ZIr/{L '7‘ My Since Q 3 T, , 1'0 , K , and r0 are all measureable, we GI “W Ih\ can easily compute the Langmuir film boundry n, and thus find the film thickness. If we divide (25) by two we obtain * Kimball and King p. 11 -- 11 _- c2 = .ézzfiZL jE;ZL. . (26) ”2“ A); a z ’3 Equation (26) shows that if half the temperature drop takes place between the hot cylinder and the Langmuir film boundry r , then in the case of pure conduction , half the heat is I. transported beyond the film. A comparison of (26) with (23) which we take as the math- ematical equivalent of the experimental law II shows that 77-2; Z /oy a: o\ (a (37) 7.5;; kin/(f {4; = I; 5’ ‘7“ -/j Thus we have the relation between the constant a and the Langmuir film thickness rL- r0. Tt is interesting to compare (27) with (5) and note that the Langmuir film thickness is the limiting value of (5) as y becomes infinite This suggests that the isothermal film thickness at the top of the cylinder is approximately the same as the Langmuir film thickness. This fact was also apparent in the case of a hot vertical plate '. Restricting the discussion now to the isothermal film as defined by (5), we shall find the total amount of heat carried away vertically from this film. The excess energy per unit volume coming from the hot cylinder is §_nk/T-7;) where k is Boltzmann's constant. Hence the flux is ‘ Kimball and King p. 11 .The heat carried away inside the film, which we have shown to be ég- is the integral of the flux over the total surface thru which heat flows, hence (28) Q = 3/74 ”(T—7:) VM/r . ’3 By use of the gas law (12) this may be written I a” m (39) 2Q- =3f/‘/9/ 7:7; V/"é/l’ = Eff/7:7; V5262 /‘ l a I The right band member of (29) is obtained by substituting for r , r and rm the values defined by (15). We now 0’ express the integrand of the right member of (29) as a power series in x up to and including terms of the third order by using the expression for v from (19) and the power series 7—7; expansions of and of 42‘ . Performing the indicated integration we obtain for the heat carried away inside the £11: ' .61...- 37/79 ”'7’: /,r 7) {(+3 ,1} j Z /é7rf'7; /7' a 33w 4"“) (30) 'h01'91n [’ = /,4_ A ,1 81", 2.2a) Jaw ’ *4 —-' I 29- /I-Z)~§——— i .. , 2’7 and. E = 2w _ a.” raw/5-5 )- /”r*?§)('77+65’)+ @213 ff” 3 27; I; be“) -- 13 -- VI. EVALUATION AND SIGNIFICANCE OF CONSTANTS To determine the constant a we equate (30) and (23) and find that ,5 a = PI." /r—7;)“ .1. wherein p: /~’2’z””j /("‘.§2‘”*«if&2) ]4' 3277:T,2k1.l